Currents and the Energy-Momentum Tensor in Classical Field Theory: A fresh look at an Old Problem

TL;DR

This paper revisits the classical definitions of currents and the energy-momentum tensor in field theory, introducing a geometric principle of ultralocality to resolve ambiguities and connect scale invariance with traceless energy-momentum tensors.

Contribution

It introduces an ultralocality principle that uniquely determines the improved currents and energy-momentum tensor in classical field theory within a geometric framework.

Findings

Ultralocality fixes ambiguity in current and energy-momentum tensor definitions.

The approach applies even when the Lagrangian depends on curvature.

Establishes link between scale invariance and traceless energy-momentum tensor.

Abstract

We give a comprehensive review of various methods to define currents and the energy-momentum tensor in classical field theory, with emphasis on a geometric point of view. The necessity of ``improving'' the expressions provided by the canonical Noether procedure is addressed and given an adequate geometric framework. The main new ingredient is the explicit formulation of a principle of ``ultralocality'' with respect to the symmetry generators, which is shown to fix the ambiguity inherent in the procedure of improvement and guide it towards a unique answer: when combined with the appropriate splitting of the fields into sectors, it leads to the well-known expressions for the current as the variational derivative of the matter field Lagrangian with respect to the gauge field and for the energy-momentum tensor as the variational derivative of the matter field Lagrangian with respect to the metric tensor. In the second case, the procedure is shown to work even when the matter field Lagrangian depends explicitly on the curvature, thus establishing the correct relation between scale invariance, in the form of local Weyl invariance ``on shell'', and tracelessness of the energy-momentum tensor, required for a consistent definition of the concept of a conformal field theory.